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Control of Fluctuations and Heavy Tails for Heat Variation in the Two-Time Measurement Framework

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Abstract

We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.

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References

  1. Andrieux, D., Gaspard, P., Monnai, T., Tasaki, S.: The fluctuation theorem for currents in open quantum systems. New J. Phys. 11(4), 043014 (2009)

    Article  ADS  Google Scholar 

  2. Arai, A.: On a model of a harmonic oscillator coupled to a quantized, massless, scalar field. I. J. Math. Phys. 22(11), 2539–2548 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Araki, H., Woods, E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas. J. Math. Phys. 4(5), 637–662 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  4. Araki, H., Wyss, W.: Representations of canonical anticommutation relations. Helv. Phys. Acta 37(2), 136 (1964)

    MathSciNet  MATH  Google Scholar 

  5. Benoist, T., Fraas, M., Jakšić, V., Pillet, C.-A.: Full statistics of erasure processes: isothermal adiabatic theory and a statistical Landauer principle. Revue Roum. Math. Pures Appl. 62(1), 259–286 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Benoist, T., Jakšić, V., Panati, A., Pautrat, Y., Pillet, C.-A.: Full statistics of energy conservation in two-time measurement protocols. Phys. Rev. E 92(3), 032115 (2015)

    Article  ADS  Google Scholar 

  7. Benoist, T., Jakšić, V., Pillet, C.-A.: Energy statistics in open harmonic networks. J. Stat. Phys. 168(5), 1016–1030 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Benoist, T., Panati, A., Pautrat, Y.: Heat conservation and fluctuation relations for open quantum systems (2018). ArXiv preprint arXiv:1810.09999

  9. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  10. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  11. Campisi, M., Blattmann, R., Kohler, S., Zueco, D., Hänggi, P.: Employing circuit QED to measure non-equilibrium work fluctuations. New J. Phys. 15(10), 105028 (2013)

    Article  ADS  Google Scholar 

  12. Campisi, M., Hänggi, P., Talkner, P.: Colloquium: Quantum fluctuation relations: foundations and applications. Rev. Mod. Phys. 83(3), 771 (2011)

    Article  ADS  MATH  Google Scholar 

  13. Chernoff, P.R., Marsden, J.E.: Some basic properties of infinite dimensional Hamiltonian systems. Colloques internationaux CNRS, Géométrie symplectique et physique mathématique 237, 313–330 (1976)

    Google Scholar 

  14. Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60(3), 2721 (1999)

    Article  ADS  Google Scholar 

  15. Davies, E.B.: The harmonic oscillator in a heat bath. Commun. Math. Phys. 33(3), 171–186 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  16. Dorner, R., Clark, S., Heaney, L., Fazio, R., Goold, J., Vedral, V.: Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry. Phys. Rev. Lett. 110(23), 230601 (2013)

    Article  ADS  Google Scholar 

  17. Dereziński, J.: Van Hove Hamiltonians–exactly solvable models of the infrared and ultraviolet problem. Ann. Henri Poincaré 4(4), 713–738 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dereziński, J.: Introduction to representations of the canonical commutation and anticommutation relations. In: Dereziński, J., Siedentop, H. (eds.) Large Coulomb Systems. Lecture Notes in Physics, vol. 695, pp. 63–143. Springer, Berlin (2006)

    Chapter  Google Scholar 

  19. Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs on Mathematical Physics. University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  20. Dereziński, J., Jakšić, V., Pillet, C.-A.: Perturbation theory of W*-dynamics, Liouvilleans and KMS-states. Rev. Math. Phys. 15(05), 447–489 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71(15), 2401 (1993)

    Article  ADS  MATH  Google Scholar 

  22. Esposito, M., Harbola, U., Mukamel, S.: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81(4), 1665 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74(14), 2694 (1995)

    Article  ADS  Google Scholar 

  24. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80(5–6), 931–970 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78(14), 2690 (1997)

    Article  ADS  Google Scholar 

  26. Jakšić, V., Kritchevski, E., Pillet, C.-A.: Mathematical theory of the Wigner–Weisskopf atom. In: Large Coulomb Systems, pp. 145–215 (2006)

  27. Jakšić, V., Ogata, Y., Pautrat, Y., Pillet, C.-A.: Entropic fluctuations in quantum statistical mechanics an introduction. In: Frohlich, J., Salmhofer, M., Mastropietro, V., De Roeck, W., Cugliandolo, L.F. (eds.) Quantum Theory from Small to Large Scales. Lecture Notes of the Les Houches Summer School, vol. 95, pp. 213–410. Oxford University Press, Oxford (2011)

    Google Scholar 

  28. Jakšić, V., Pillet, C.-A.: Ergodic properties of classical dissipative systems I. Acta Math. 181(2), 245–282 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jakšić, V., Panangaden, J., Panati, A., Pillet, C.-A.: Energy conservation, counting statistics, and return to equilibrium. Lett. Math. Phys. 105(7), 917–938 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Jakšić, V., Pillet, C.-A., Rey-Bellet, L.: Entropic fluctuations in statistical mechanics: I. Classical dynamical systems. Nonlinearity 24(3), 699 (2011)

    MATH  Google Scholar 

  31. Jakšić, V., Pillet, C.-A., Shirikyan, A.: Entropic fluctuations in Gaussian dynamical systems. Rep. Math. Phys. 77(3), 335–376 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Kurchan, J.: A quantum fluctuation theorem (2000). ArXiv preprint arXiv:cond-mat/0007360

  33. Mazzola, L., De Chiara, G., Paternostro, M.: Measuring the characteristic function of the work distribution. Phys. Rev. Lett. 110(23), 230602 (2013)

    Article  ADS  Google Scholar 

  34. Raquépas, R.: Heat full statistics and regularity of perturbations in quantum statistical mechanics. Master’s thesis. McGill University, Montréal (2017)

  35. Roncaglia, A.J., Cerisola, F., Paz, J.P.: Work measurement as a generalized quantum measurement. Phys. Rev. Lett. 113(25), 250601 (2014)

    Article  ADS  Google Scholar 

  36. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I: Functional Analysis. Academic Press, New York (1972)

  37. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

  38. Spohn, H., Lebowitz, J.L.: Stationary non-equilibrium states of infinite harmonic systems. Commun. Math. Phys. 54(2), 97–120 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  39. Spohn, H.: Dynamics of Charged Particles and Their Radiation Field. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  40. Tasaki, H.: Jarzynski relations for quantum systems and some applications (2000). ArXiv preprint arXiv:cond-mat/0009244

  41. Thirring, W.: A Course in Mathematical Physics 1 and 2: Classical Dynamical Systems and Classical Field Theory. Springer Science & Business Media, Berlin (2012)

    Book  Google Scholar 

  42. Talkner, P., Lutz, E., Hänggi, P.: Fluctuation theorems: work is not an observable. Phys. Rev. E 75, 050102 (2007)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Institut de mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, 31 062, Toulouse Cedex 9, France

    Tristan Benoist

  2. Aix Marseille Univ, Université de Toulon CNRS, CPT, Marseille, France

    Annalisa Panati

  3. Department of Mathematics and Statistics, McGill University, 1005-805 rue Sherbrooke Ouest, Montréal, QC  , H3A 0B9, Canada

    Renaud Raquépas

  4. Univ. Grenoble Alpes, CNRS, Institut Fourier, 38 000, Grenoble, France

    Renaud Raquépas

Authors
  1. Tristan Benoist
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  2. Annalisa Panati
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  3. Renaud Raquépas
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Correspondence to Annalisa Panati.

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Communicated by Christian Maes.

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Benoist, T., Panati, A. & Raquépas, R. Control of Fluctuations and Heavy Tails for Heat Variation in the Two-Time Measurement Framework. Ann. Henri Poincaré 20, 631–674 (2019). https://doi.org/10.1007/s00023-018-0743-x

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  • DOI: https://doi.org/10.1007/s00023-018-0743-x

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